When in battle, Cyclops wears his visor for channelling his Optic Energy Beams.
He uses the visor to control his Optic beams. Without the visor he can't open his eyes without emitting a destructive optic blast from his eyes. When trying to fit in and during the time he isn't fighting as an X-Man he wears sunglasses made of a similar material to the visor that prevents his optic blasts from destroying everything in sight.
In the second film the visor has a more sleek, slim look and no longer covers his ears. This has no other obvious effect other than looking more compact, he still maintains the same amount of control over the beams using the slimmer visor.